Journal
MATHEMATICS OF COMPUTATION
Volume 78, Issue 268, Pages 2209-2222Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0025-5718-09-02217-0
Keywords
Diophantine approximation; multidimensional continued fraction algorithm
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It has been believed that the continued fraction expansion of (alpha, beta) (1, alpha, beta is a Q-basis of a real cubic field) obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of (<(3)root 3 >, <(3)root 3 >) (< x > denoting the fractional part of x). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of (alpha, beta) with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.
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